Quantum Tomography under Prior Information

By Teiko Heinosaari, Luca Mazzarella, Michael M. Wolf. 2011

212 citations recorded by the source corpus.

We provide a detailed analysis of the question: how many measurement settings or outcomes are needed in order to identify an unknown quantum state which is constrained by prior information? We show that if the prior information restricts the possible states to a set of lower dimensionality, then topological obstructions can increase the required number of outcomes by a factor of two over the number of real parameters needed to characterize the set of all states. Conversely, we show that almost every measurement becomes informationally complete with respect to the constrained set if the number of outcomes exceeds twice the Minkowski dimension of the set. We apply the obtained results to determine the minimal number of outcomes of measurements which are informationally complete with respect to states with rank constraints. In particular, we show that the minimal number of measurement outcomes (POVM elements) necessary to identify all pure states in a d-dimensional Hilbert space is 4d−3−c(d) α(d) for some $${c(d)\in[1,2]}$$ and α(d) being the number of ones appearing in the binary expansion of (d−1).

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/bac3c286a1ac8d02158234add9cef874ed22572b

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In Atomic and Molecular Physics, and Optics

Citations in the corpus

Compressive phase retrieval via generalized approximate message passing (2012, background): “To reconstruct (up to a global phase uncertainty), it has been recently established that intensity measurements are necessary [15] and are sufficient [16] through appropriate design of the linear transform .”
Phase Retrieval from Coded Diffraction Patterns (2013, unknown): “Other areas where related problems appear include acoustics [12, 8], blind channel estimation in wireless communications [4, 45], interferometry [28], quantum mechanics [25, 47] and quantum information [34].”
Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators (2012, background): “While this intuition is indeed correct [20– 23], it is a challenge to devise a practical tomography scheme that takes advantage of this.”
Phase Retrieval via Matrix Completion (2011, unknown): “[36].”
A Geometric Analysis of Phase Retrieval (2016, background): “GPR also sees applications in electron microscopy [71], diffraction and array imaging [26,34], acoustics [15,16], quantum mechanics [38,86] and quantum information [54].”
Phase Retrieval via Wirtinger Flow: Theory and Algorithms (2014, methodology): “In this setup, the most widely used method is perhaps the error reduction algorithm and its generalizations, which were derived from the pioneering research of Gerchberg and Saxton [29] and Fienup [24,26].”